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Theorem findabrcl 29524
Description: Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
findabrcl.1  |-  ( z  e.  P  ->  ( G `  z )  e.  P )
Assertion
Ref Expression
findabrcl  |-  ( ( C  e.  om  /\  A  e.  P )  ->  ( ( x  e. 
_V  |->  ( rec ( G ,  A ) `  x ) ) `  C )  e.  P
)
Distinct variable groups:    x, G    x, A    x, C    z, G    z, A    z, P
Allowed substitution hints:    C( z)    P( x)

Proof of Theorem findabrcl
StepHypRef Expression
1 elex 3122 . . . 4  |-  ( C  e.  om  ->  C  e.  _V )
2 fveq2 5866 . . . . 5  |-  ( x  =  C  ->  ( rec ( G ,  A
) `  x )  =  ( rec ( G ,  A ) `  C ) )
3 eqid 2467 . . . . 5  |-  ( x  e.  _V  |->  ( rec ( G ,  A
) `  x )
)  =  ( x  e.  _V  |->  ( rec ( G ,  A
) `  x )
)
4 fvex 5876 . . . . 5  |-  ( rec ( G ,  A
) `  C )  e.  _V
52, 3, 4fvmpt 5950 . . . 4  |-  ( C  e.  _V  ->  (
( x  e.  _V  |->  ( rec ( G ,  A ) `  x
) ) `  C
)  =  ( rec ( G ,  A
) `  C )
)
61, 5syl 16 . . 3  |-  ( C  e.  om  ->  (
( x  e.  _V  |->  ( rec ( G ,  A ) `  x
) ) `  C
)  =  ( rec ( G ,  A
) `  C )
)
76adantr 465 . 2  |-  ( ( C  e.  om  /\  A  e.  P )  ->  ( ( x  e. 
_V  |->  ( rec ( G ,  A ) `  x ) ) `  C )  =  ( rec ( G ,  A ) `  C
) )
8 findabrcl.1 . . . 4  |-  ( z  e.  P  ->  ( G `  z )  e.  P )
98findreccl 29523 . . 3  |-  ( C  e.  om  ->  ( A  e.  P  ->  ( rec ( G ,  A ) `  C
)  e.  P ) )
109imp 429 . 2  |-  ( ( C  e.  om  /\  A  e.  P )  ->  ( rec ( G ,  A ) `  C )  e.  P
)
117, 10eqeltrd 2555 1  |-  ( ( C  e.  om  /\  A  e.  P )  ->  ( ( x  e. 
_V  |->  ( rec ( G ,  A ) `  x ) ) `  C )  e.  P
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    |-> cmpt 4505   ` cfv 5588   omcom 6684   reccrdg 7075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6685  df-recs 7042  df-rdg 7076
This theorem is referenced by: (None)
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